Question

Eggs with a mass of \(0.15 \mathrm{~kg}\) per egg and a specific heat of \(3.32 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}\) are cooled from \(32^{\circ} \mathrm{C}\) to \(10^{\circ} \mathrm{C}\) at a rate of 200 eggs per minute. The rate of heat removal from the eggs is (a) \(7.3 \mathrm{~kW}\) (b) \(53 \mathrm{~kW}\) (c) \(17 \mathrm{~kW}\) (d) \(438 \mathrm{~kW}\) (e) \(37 \mathrm{~kW}\)

Short answer

Question: The eggs have a mass of 0.15 kg and specific heat of 3.32 kJ/kg°C. They are cooled down from 32°C to 10°C at a rate of 200 eggs per minute. Calculate the rate of heat removal from the eggs. Answer: The rate of heat removal from the eggs is approximately 36.38 kW.

Step-by-step solution

Calculate the heat removed from one egg

Calculate the heat removed from one egg using the formula: Q = mcΔT where Q is the heat removed, m is the mass of the egg (0.15 kg), c is the specific heat capacity (3.32 kJ/kg°C), and ΔT is the change in temperature (from 32°C to 10°C). ΔT = 32 - 10 = 22°C Q = (0.15 kg) × (3.32 kJ/kg°C) × (22°C) = 10.914 kJ

Convert the heat removed into watts

We need to convert the heat removed per egg into watts. 1 kJ = 1000 J, and since there are 60 seconds in a minute, we need to divide the value by 60 to get the heat removed per second: Q = (10.914 kJ) × (1000 J/kJ) × (1/60) = 181.9 J/s = 181.9 W

Calculate the heat removed for 200 eggs per minute

Now, we need to calculate the heat removed for all the eggs cooled per minute. Since there are 200 eggs cooled per minute, we can simply multiply the heat removed per egg with the number of eggs: Total heat removed = (181.9 W) × (200 eggs) = 36380 W

Convert the heat removed into kilowatts

Finally, we need to convert the total heat removed per minute into kilowatts: Total heat removed = 36380 W = 36.38 kW The closest answer in the given options is (e) 37 kW.

Key concepts

Heat Transfer

Heat transfer is an important concept in thermodynamics that involves the exchange of thermal energy between physical systems. In this exercise, we are looking at how heat is transferred away from eggs during a cooling process.
There are three main modes of heat transfer:

• Conduction: Transfer of heat through a material without the movement of the material itself. This involves particles passing on their kinetic energy to neighboring particles.
• Convection: Transfer of heat by the movement of fluids (liquids or gases). This occurs when hotter, less dense areas of a fluid rise, while cooler, denser areas sink, creating a current.
• Radiation: Transfer of energy by electromagnetic waves. This does not require a medium, so it can occur through a vacuum, like the heat from the sun.

In the problem described, the cooling of eggs primarily involves conduction and possibly convection. Heat is removed from the eggs into the surrounding environment, reducing the eggs' temperature. This removal must be in balance with the rate of cooling to efficiently lower the temperature from 32°C to 10°C.

Specific Heat Capacity

The specific heat capacity is a property that indicates how much heat energy is needed to change the temperature of a substance by a certain amount. It is the amount of heat required to raise the temperature of 1 kg of a substance by 1°C. In our exercise, specific heat capacity is crucial in calculating the heat removed from eggs during cooling.
The formula used is:\[ Q = mc\Delta T \]where:

• \( Q \) is the heat energy transferred (in joules or kilojoules),
• \( m \) is the mass of the substance (in kilograms),
• \( c \) is the specific heat capacity (in kJ/kg°C), and
• \( \Delta T \) is the change in temperature (in °C).

In this scenario, the specific heat capacity of the eggs is provided as 3.32 kJ/kg°C. This value helps determine how effectively the eggs absorb or lose heat as they cool down, by allowing us to calculate the total energy removed per egg through the expression \( 10.914 \) kJ. The high specific heat value suggests that eggs can hold a significant amount of heat, necessitating a noticeable cooling effort to change their internal temperatures significantly.

Cooling Rate

Cooling rate refers to how quickly an object's temperature decreases over time. In the context of this exercise, it pertains to the rate at which heat is removed from the eggs to lower their temperature from 32°C to 10°C efficiently. A faster cooling rate implies a greater rate of heat removal.
Here's how it works in this case:- The cooling rate is determined by calculating how much heat needs to be removed to achieve the desired temperature drop.- Once we know the heat energy removed from a single egg (10.914 kJ), we multiply it by the cooling rate, which is the number of eggs cooled per minute (200 eggs).- By converting this energy to watts (power),\[ 181.9 \text{ W (per egg)} \times 200 = 36,380 \text{ W} \], we get the total power requirement for the cooling process, which is approximately 36.38 kW.
Understanding the cooling rate is vital for designing efficient cooling systems. It allows engineers to ensure that enough energy is removed from a system to maintain desired temperatures in processes involving large numbers of items, like our batch of eggs in the problem. Adjusting factors such as the cooling method or environmental conditions can also optimize the cooling rate further.